Related papers: On planar Beltrami equations and Hoelder regularit…
We estimate the Hoelder exponent $\alpha$ of solutions to the Beltrami equation $\dbar f=\mu\de f$, where the Beltrami coefficient satisfies $\|\mu\|_\infty<1$. Our estimate improves the classical estimate $\alpha\ge\|K_\mu\|^{-1}$, where…
New results regarding the Sobolev regularity of the principal solution of the linear Beltrami equation $\bar{\partial} f = \mu \partial f + \nu \overline{\partial f}$ for discontinuous Beltrami coefficients $\mu$ and $\nu$ are obtained,…
We establish surprising improved Schauder regularity properties for solutions to the Leray-Lions divergence type equation in the plane. The results are achieved by studying the nonlinear Beltrami equation and making use of special new…
Our goal in this work is to present some function spaces on the complex plane $\C$, $X(\C)$, for which the quasiregular solutions of the Beltrami equation, $\bar\partial f (z) = \mu(z) \partial f (z)$, have first derivatives locally in…
We establish a series of criteria on the existence of regular solutions for the Dirichlet problem to general degenerate Beltrami equations ${\bar{\partial}}f = \mu {\partial f}+\nu {\bar{\partial f}}$ in arbitrary Jordan domains in $\C$.
In this article, first we give a general lemma on the existence of regular homeomorphic solutions $f$ with the hydrodynamic normalization $f(z)=z+o(1)$ as $z\to\infty$ to the degenerate Beltrami equations $\overline{\partial}f=\mu\,\partial…
We study Hardy spaces $H^p_\nu$ of the conjugate Beltrami equation $\bar{\partial} f=\nu\bar{\partial f}$ over Dini-smooth finitely connected domains, for real contractive $\nu\in W^{1,r}$ with $r>2$, in the range $r/(r-1)<p<\infty$. We…
We consider problems concerning the existence of solutions of the Beltrami equations and their convergence in the entire complex plane. We are mainly interested in the case when these solutions satisfy the so-called hydrodynamic…
We study the distributional solutions to the (generalized) Beltrami equation under Sobolev assumptions on the Beltrami coefficients. In this setting, we prove that these distributional solutions are true quasiregular maps and they are…
An effective algorithm is presented for solving the Beltrami equation fzbar = mu fz in a planar disk. The algorithm involves no evaluation of singular integrals. The strategy, working in concentric rings, is to construct a piecewise linear…
This article addresses the planar Coleman--Gurtin heat equation with memory on a bounded domain, with rough anisotropic diffusion $A_\mu$, typical of heterogeneous or composite media and encoded by a Beltrami coefficient $\mu\in…
In this paper we extend the interior regularity results for stable solutions in [Cabr\'{e}, Figalli, Ros-Oton, and Serra, Acta Math. 224 (2020)] to operators with variable coefficients. We show that stable solutions to the semilinear…
An effective algorithm is presented for solving the Beltrami equation df/dz = mu (df/dzbar) in a planar disk. The disk is triangulated in a simple way and f is approximated by piecewise linear mappings; the images of the vertices of the…
We study mappings with branching of a domain of Euclidean space. The H\"older and Lipschitz continuity are established for one class of spatial mappings whose characteristic satisfies the Dini type condition in a given domain. In addition,…
We prove a surprising higher regularity for solutions to the nonlinear elliptic autonomous Beltrami equation in a planar domain $\Omega$, \[ f_\zbar = {\cal A}(f_z) \hskip15pt a.e.\;\; z\in \Omega, \] when ${\cal A}$ is linear at $\infty$.…
We study some size estimates for the solution of the equation d-bar u=f in one variable. The new ingredient is the use of holomorphic functions with precise growth restrictions in the construction of explicit solutions to the equation.
In this paper we show that the homeomorphic solutions to each nonlinear Beltrami equation $\partial_{\bar{z}} f = \mathcal{H}(z, \partial_{z} f)$ generate a two-dimensional manifold of quasiconformal mappings $\mathcal{F}_{\mathcal{H}}…
We study Hardy spaces of solutions to the conjugate Beltrami equation with Lipschitz coefficient on Dini-smooth simply connected planar domains, in the range of exponents $1<\infty$. We analyse their boundary behaviour and certain density…
The paper concerns numerical algorithms for solving the Beltrami equation $f_{\bar{z}}(z)=\mu(z) f_z(z)$ for a compactly supported $\mu$. First, we study an efficient algorithm that has been proposed in the literature, and present its…
As in the work of Tartar ( Tartar L. Interpolation non lin\'eaire et r\'egularit\'e, 9, Journal of Functional Analysis, (1972), 469-489) we developed here some new results on non linear interpolation of $\alpha$-H\"olderian mappings between…